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Chapter 1: Problem 21

Find the domain of each function. $$ \frac{\sqrt{x+2}}{x} $$

Short Answer

Expert verified
The domain is \(( -2, 0 ) \cup ( 0, \, \infty )\).

Step by step solution

01

Identify the Conditions for the Square Root

The expression under the square root must be non-negative for the square root to be defined. Therefore, we need to ensure that \( x + 2 \geq 0 \). This simplifies to \( x \geq -2 \).
02

Determine Restrictions for the Denominator

Since division by zero is undefined, the denominator of the function, \( x \), cannot be zero. Therefore, \( x eq 0 \).
03

Combine the Conditions

Combine the two conditions from Step 1 and Step 2. The domain of the function is all real numbers where \( x \geq -2 \) and \( x eq 0 \).
04

Express the Domain in Interval Notation

The domain can be expressed in interval notation as \(( -2, 0 ) \cup ( 0, \, \infty )\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Square Root Function
The square root function is a fundamental concept in algebra that involves finding the principal square root of a number. When dealing with a function that includes a square root, such as \( \sqrt{x+2} \), it is crucial to consider its domain. The domain of a square root function is the set of all real numbers for which the expression under the square root is non-negative. This is important because we cannot take the square root of a negative number in the set of real numbers.
  • For \( \sqrt{x+2} \), we need \( x+2 \geq 0 \) to ensure that the expression remains defined on the real number set.
  • This condition tells us that \( x \geq -2 \). These are the permissible values for \( x \) that keep our function valid.
Remember that while the square root function can accept zero since \( \sqrt{0} = 0 \), all values below the threshold (-2 in this case) would result in an undefined operation in the real numbers.
Interval Notation Simplified
When describing the domain of a function, interval notation provides a concise and clear way to represent a set of numbers. It uses parentheses and brackets to define a span of values.
  • In interval notation, square brackets \([ ]\) indicate that an endpoint is included in the interval, corresponding to \( \leq \) or \( \geq \).
  • Parentheses \(( )\), on the other hand, mean the endpoint is not included, corresponding to '<' or '>' in inequalities.
For our function \( \frac{\sqrt{x+2}}{x} \), after determining that \( x \geq -2 \) and \( x eq 0 \), we can use interval notation to express this domain. Since \( x eq 0 \), zero is not included in the domain, leading to the representation of \(( -2, 0 ) \cup ( 0, \infty )\). This shows two separate intervals where the function is defined.
Avoiding Division by Zero
Division by zero is a specific situation in mathematics that must be avoided because it leads to undefined results. In a rational function, the denominator cannot be zero, or the expression becomes impossible to evaluate meaningfully.
  • For a simple fraction like \( \frac{a}{b} \), if \( b = 0 \), the fraction is undefined.
  • This is true for any function, such as \( \frac{\sqrt{x+2}}{x} \), where we must ensure the denominator \( x eq 0 \).
Understanding why division by zero cannot be allowed helps identify the values that must be excluded from the domain of a function. In our example, zero is a critical exclusion that ensures the function remains defined and computationally viable.

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Most popular questions from this chapter

If an object is initially at a height above the ground of \(s_{0}\) feet and is thrown straight upward with an initial velocity of \(v_{0}\) feet per second, then from physics it can be shown the height in feet above the ground is given by \(s(t)=-16 t^{2}+v_{0} t+s_{0},\) where \(t\) is in seconds. Find how long it takes for the object to reach maximum height. Find when the object hits the ground.

Dowdy \(^{52}\) found that the percentage \(y\) of the lesser grain borer beetle initiating flight was approximated by the equation \(y=-240.03+17.83 T-0.29 T^{2}\) where \(T\) is the temperature in degrees Celsius. a. Find the temperature at which the highest percentage of beetles would fly. b. Find the minimum temperature at which this beetle initiates flight. c. Find the maximum temperature at which this beetle initiates flight.

Pine and Allen \(^{89}\) studied the growth habits of sturgeon in the Suwannee River, Florida. The sturgeon fishery was once an important commercial fishery but, because of overfishing, was closed down in \(1984,\) and the sturgeon is now both state and federally protected. The mathematical model that Pine and Allen created was given by the equation \(L(t)=222.273(1-\) \(\left.e^{-0.08042[t+2.181}\right),\) where \(t\) is age in years and \(L\) is length in centimeters. Graph this equation. Find the expected age of a 100 -cm-long sturgeon algebraically.

If you use the quadratic function \(C(x)=\) \(a x^{2}+b x+c\) to model costs on a very large interval, what sign should the coefficient \(a\) have? Explain carefully.

Using your grapher, graph on the same screen \(y_{1}=\) \(f(x)=|x|\) and \(y_{2}=g(x)=\sqrt{x^{2}} .\) How many graphs do you see? What does this say about the two functions?
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